Properties

Label 2.0.7.1-6300.2-c5
Base field \(\Q(\sqrt{-7}) \)
Conductor \((-60 a + 30)\)
Conductor norm \( 6300 \)
CM no
Base change yes: 210.e7,1470.j7
Q-curve yes
Torsion order \( 16 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-7}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 2 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 2)
 
gp: K = nfinit(a^2 - a + 2);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
 

Weierstrass equation

\(y^2+xy=x^{3}+210x+900\)
sage: E = EllipticCurve(K, [1, 0, 0, 210, 900])
 
gp: E = ellinit([1, 0, 0, 210, 900],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 0, 210, 900]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-60 a + 30)\) = \( \left(a\right) \cdot \left(-a + 1\right) \cdot \left(3\right) \cdot \left(5\right) \cdot \left(-2 a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 6300 \) = \( 2^{2} \cdot 7 \cdot 9 \cdot 25 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((928972800)\) = \( \left(a\right)^{16} \cdot \left(-a + 1\right)^{16} \cdot \left(3\right)^{4} \cdot \left(5\right)^{2} \cdot \left(-2 a + 1\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 862990463139840000 \) = \( 2^{32} \cdot 7^{2} \cdot 9^{4} \cdot 25^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1023887723039}{928972800} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(6 a - 6 : -18 a - 24 : 1\right)$
Height \(0.621045073666089\)
Torsion structure: \(\Z/2\Z\times\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-12 a : 66 : 1\right)$ $\left(-\frac{45}{4} a + \frac{15}{2} : \frac{45}{8} a - \frac{15}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.621045073666089 \)
Period: \( 0.442077482916679 \)
Tamagawa product: \( 4096 \)  =  \(2^{4}\cdot2^{4}\cdot2\cdot2^{2}\cdot2\)
Torsion order: \(16\)
Leading coefficient: \(6.64129038694595\)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a\right) \) \(2\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)
\( \left(-a + 1\right) \) \(2\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)
\( \left(-2 a + 1\right) \) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(5\right) \) \(25\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 6300.2-c consists of curves linked by isogenies of degrees dividing 32.

Base change

This curve is the base change of elliptic curves 210.e7, 1470.j7, defined over \(\Q\), so it is also a \(\Q\)-curve.